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Theorem cocanfo 37726
Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
cocanfo (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 769 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺𝐹) = (𝐻𝐹))
21fveq1d 6908 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = ((𝐻𝐹)‘𝑦))
3 simpl1 1192 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴onto𝐵)
4 fof 6820 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
53, 4syl 17 . . . . . 6 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴𝐵)
6 fvco3 7008 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
75, 6sylan 580 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
8 fvco3 7008 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
95, 8sylan 580 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
102, 7, 93eqtr3d 2785 . . . 4 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
1110ralrimiva 3146 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
12 fveq2 6906 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐺‘(𝐹𝑦)) = (𝐺𝑥))
13 fveq2 6906 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
1412, 13eqeq12d 2753 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ (𝐺𝑥) = (𝐻𝑥)))
1514cbvfo 7309 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
163, 15syl 17 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
1711, 16mpbid 232 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥))
18 eqfnfv 7051 . . . 4 ((𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
19183adant1 1131 . . 3 ((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2019adantr 480 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2117, 20mpbird 257 1 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  ccom 5689   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569
This theorem is referenced by: (None)
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